Search Previous Year Questions

Shortcut Trick

Solve the bracket first: (36 + 24) = 60.

Group positive and negative terms: (78 + 69 + 85 + 32) − (60 + 49).

Simplify: 264 − 109 = 155.

∴ The correct answer is 155.

Alternate Method

Given: 78 + 69 − (36 + 24) + 85 − 49 + 32

Method Used: Apply the BODMAS rule (Brackets, Orders, Division, Multiplication, Addition, Subtraction).

⇒ 78 + 69 − 60 + 85 − 49 + 32

⇒ 147 − 60 + 85 − 49 + 32

⇒ 87 + 85 − 49 + 32

⇒ 172 − 49 + 32

⇒ 123 + 32

⇒ 155

∴ The correct answer is 155.

Additional Information

BODMAS Rule

The BODMAS rule is an acronym to help remember the order of operations in calculations: Brackets, Orders, Division, Multiplication, Addition, and Subtraction.

Arithmetic Signs

When simplifying expressions with multiple signs, always process terms inside brackets first. For addition and subtraction, operations can be performed from left to right or by grouping like signs together.

Shortcut Trick

Solve the operations using BODMAS order: Of → Division → Addition/Subtraction.

First, calculate Of: 18 × 2 = 36 and 13 × 2 = 26.

Then, calculate Division: 36 ÷ 3 = 12.

Combine results: 59 + 36 − 12 + 44 − 21 + 26 = 132.

∴ The correct answer is 132.

Alternate Method

Given Expression: 59 + (18 of 2) − 36 ÷ 3 + 44 − 21 + 13 of 2

Method Used: Apply the BODMAS rule (Bracket, Of, Division, Multiplication, Addition, Subtraction).

⇒ 59 + (18 × 2) − 36 ÷ 3 + 44 − 21 + (13 × 2)

⇒ 59 + 36 − 36 ÷ 3 + 44 − 21 + 26

⇒ 59 + 36 − 12 + 44 − 21 + 26

⇒ (59 + 36 + 44 + 26) − (12 + 21)

⇒ 165 − 33

⇒ 132

∴ The correct answer is 132.

Additional Information

BODMAS / VBODMAS Rule

This rule defines the priority of arithmetic operations: V (Vinculum/Bar), B (Brackets), O (Of/Orders), D (Division), M (Multiplication), A (Addition), and S (Subtraction).

The 'Of' Operator

In mathematical simplification, Of is treated as a multiplication that must be performed after brackets but strictly before division and multiplication.

Grouping Strategy

To minimize errors in long expressions, group all positive terms and negative terms separately before performing the final subtraction.

Shortcut Trick

Solve the numerator directly: 0.4 × 0.4 + 0.02 = 0.16 + 0.02 = 0.18.

Solve the denominator directly: 0.3 × 0.3 + 0.03 = 0.09 + 0.03 = 0.12.

Divide the results: 0.18 ÷ 0.12 = 18 ÷ 12 = 1.5.

∴ The correct answer is 1.5.

Alternate Method

Given:

Expression = (0.4 × 0.4 + 0.02) ÷ (0.3 × 0.3 + 0.03)

Method: Apply the BODMAS rule, solving brackets and multiplication before addition and division.

⇒ Numerator = 0.4 × 0.4 + 0.02

⇒ Numerator = 0.16 + 0.02 = 0.18

⇒ Denominator = 0.3 × 0.3 + 0.03

⇒ Denominator = 0.09 + 0.03 = 0.12

⇒ Expression = 0.18 ÷ 0.12

⇒ Expression = 18/12

⇒ Expression = 3/2 = 1.5

∴ The correct answer is 1.5.

Additional Information

BODMAS Rule

The order of operations is Brackets, Orders, Division, Multiplication, Addition, and Subtraction.

Decimal Multiplication

When multiplying decimals, the number of decimal places in the product is the sum of the decimal places in the values being multiplied (e.g., 0.4 × 0.4 = 0.16).

Simplifying Fractions

To simplify 0.18 ÷ 0.12, multiply both numerator and denominator by 100 to remove decimals, resulting in 18 ÷ 12.

Shortcut Trick

Identify the expression as the standard algebraic identity: ((a + b)² − (a − b)²) ÷ ab.

Where a = 20.8 and b = 1.2.

Since (a + b)² − (a − b)² = 4ab, the entire expression simplifies directly to 4ab ÷ ab = 4.

The result is independent of the actual values of a and b.

∴ The correct answer is 4.

Alternate Method

Given:

First value (a) = 20.8

Second value (b) = 1.2

Formula Used:

Expression = ((a + b)² − (a − b)²) ÷ (a × b)

Calculations:

⇒ Sum (a + b) = 20.8 + 1.2 = 22

⇒ Difference (a − b) = 20.8 − 1.2 = 19.6

⇒ Numerator = 22² − 19.6²

⇒ Numerator = 484 − 384.16 = 99.84

⇒ Denominator = 20.8 × 1.2 = 24.96

⇒ Final Value = 99.84 ÷ 24.96

⇒ Final Value = 4

∴ The correct answer is 4.

Additional Information

Square of Sum and Difference Identity

The difference between the square of a sum and the square of a difference is always four times the product: (a + b)² − (a − b)² = 4ab.

Square of Sum and Difference Addition

The sum of the square of a sum and the square of a difference is twice the sum of their squares: (a + b)² + (a − b)² = 2(a² + b²).

Difference of Two Squares

The difference of two squares is equal to the product of their sum and difference: a² − b² = (a + b)(a − b).

Shortcut Trick

Use the algebraic identity: a² − b² = (a + b)(a − b).

Let a = 81.58 and b = 29.42.

Sum (a + b) = 81.58 + 29.42 = 111

Difference (a − b) = 81.58 − 29.42 = 52.16

Product = 111 × 52.16 = 5789.76

∴ The correct answer is 5789.76.

Alternate Method

Given Expression: (81.58)² − (29.42)²

Formula Used: a² − b² = (a + b)(a − b)

⇒ a = 81.58, b = 29.42

⇒ (81.58 + 29.42) × (81.58 − 29.42)

⇒ 111 × 52.16

⇒ 111 × 52.16 = (100 + 10 + 1) × 52.16

⇒ 5216 + 521.6 + 52.16

⇒ 5789.76

∴ The correct answer is 5789.76.

Additional Information

Algebraic Identities

Standard identities like (a + b)² = a² + 2ab + b² and (a − b)² = a² − 2ab + b² are useful for simplifying squares of decimals.

Multiplication by 111

Multiplying any number by 111 is equivalent to (100 + 10 + 1) × Number, which allows for quick addition of shifted values.

Difference of Squares

The identity a² − b² = (a + b)(a − b) is most efficient when the sum or difference results in a round number like 100 or 111.

Shortcut Trick

The expression follows the algebraic identity a² + b² + 2ab = (a + b)².

Identify a = 320 and b = 180.

Direct calculation: (320 + 180)² = 500².

500² = 250000.

∴ The correct answer is 250000.

Alternate Method

Given:

Expression: (320 × 320 + 180 × 180 + 2 × 320 × 180)

Formula Used:

(a + b)² = a² + b² + 2ab

Calculations:

⇒ Let a = 320 and b = 180

⇒ The given expression is in the form: a² + b² + 2ab

⇒ Substituting the values: (320)² + (180)² + 2(320)(180)

⇒ According to the identity: (320 + 180)²

⇒ (500)²

⇒ 500 × 500 = 250000

∴ The correct answer is 250000.

Additional Information

Algebraic Identity for (a − b)²

The square of a difference is calculated as a² + b² − 2ab = (a − b)².

Difference of Two Squares

The identity a² − b² = (a + b)(a − b) is useful for simplifying the product of a sum and a difference.

Identity for (a + b + c)²

For three terms, the square is given by (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca.

Shortcut Trick

Side of the rhombus = 120 ÷ 4 = 30 cm.

Half of the given diagonal = 36 ÷ 2 = 18 cm.

Using the Pythagorean triplet (3, 4, 5) scaled by 6: (18, 24, 30).

The other half-diagonal must be 24 cm.

Total length of the other diagonal = 24 × 2 = 48 cm.

∴ The correct answer is 48 cm.

Alternate Method

Given: Perimeter = 120 cm, Diagonal 1 (d₁) = 36 cm.

Formula Used: In a rhombus, the side (s) and diagonals (d₁, d₂) are related by: s² = (d₁/2)² + (d₂/2)².

⇒ Side (s) = Perimeter ÷ 4 = 120 ÷ 4 = 30 cm

⇒ Half-diagonal₁ = 36 ÷ 2 = 18 cm

⇒ 30² = 18² + (d₂/2)²

⇒ 900 = 324 + (d₂/2)²

⇒ (d₂/2)² = 900 − 324 = 576

⇒ d₂/2 = √576 = 24 cm

⇒ Diagonal 2 (d₂) = 24 × 2 = 48 cm

∴ The correct answer is 48 cm.

Additional Information

Area of a Rhombus

The Area is calculated as 1/2 × d₁ × d₂. In this case, Area = 1/2 × 36 × 48 = 864 cm².

Properties of Diagonals

The diagonals of a rhombus bisect each other at right angles (90°) and also bisect the internal angles of the rhombus.

Side-Diagonal Relationship

The sum of the squares of the diagonals is equal to four times the square of the side: d₁² + d₂² = 4s².

Shortcut Trick

In a trapezium, the Area can be directly calculated using the median (the line joining the mid-points of non-parallel sides) and the height.

Area = Median × Height

Given: Median = 12 cm and Height = 30 cm

Area = 12 × 30 = 360 cm²

∴ The correct answer is 360 cm².

Alternate Method

Given:

Distance between parallel sides (Height, h) = 30 cm

Length of the line joining mid-points (Median, m) = 12 cm

Formula Used:

Area of a trapezium = 1/2 × (Sum of parallel sides) × Height

Median (m) = (Sum of parallel sides) ÷ 2

Calculations:

⇒ Let the parallel sides be a and b.

⇒ m = (a + b) ÷ 2 = 12 cm

⇒ Sum of parallel sides (a + b) = 12 × 2 = 24 cm

⇒ Area = 1/2 × (24) × 30

⇒ Area = 12 × 30

⇒ Area = 360 cm²

∴ The correct answer is 360 cm².

Additional Information

Median of a Trapezium

The line segment connecting the mid-points of the non-parallel sides is parallel to the bases and its length is the arithmetic mean of the bases: m = (a + b) ÷ 2.

Isosceles Trapezium

A trapezium where the non-parallel sides are equal in length and the base angles are equal. Its diagonals are also equal in length.

Distance Formula for Area

If a trapezium is placed on a coordinate plane, the area can also be found using the vertical distance (difference in y-coordinates) between the parallel bases.

Shortcut Trick

To find the Highest Common Factor (HCF) of decimals, first convert them into integers by multiplying with a power of 10.

Multiply 0.48, 0.64, and 0.80 by 100 to get 48, 64, and 80.

The common factor for 48, 64, and 80 is 16 (since 48 = 16 × 3, 64 = 16 × 4, and 80 = 16 × 5).

Divide the result back by 100: 16 ÷ 100 = 0.16.

∴ The correct answer is 0.16.

Alternate Method

Given: Numbers 0.48, 0.64, and 0.80.

Formula Used: HCF of Fractions = (HCF of Numerators) ÷ (LCM of Denominators)

⇒ Convert decimals to fractions: 0.48 = 48/100, 0.64 = 64/100, 0.80 = 80/100

⇒ Find prime factors of numerators:

⇒ 48 = 2⁴ × 3

⇒ 64 = 2⁶

⇒ 80 = 2⁴ × 5

⇒ HCF of (48, 64, 80) = 2⁴ = 16

⇒ LCM of denominators (100, 100, 100) = 100

⇒ Required HCF = 16 / 100 = 0.16

∴ The correct answer is 0.16.

Additional Information

HCF of Fractions

The Highest Common Factor of a set of fractions is calculated as the HCF of the numerators divided by the LCM of the denominators.

LCM of Decimals

To find the Least Common Multiple, make the number of decimal places equal, find the LCM of the resulting integers, and then shift the decimal point back.

Relationship Rule

For any two numbers A and B, the product of the numbers is equal to the product of their HCF and LCM: A × B = HCF × LCM.

Shortcut Trick

Identify the largest number in the set: 20.

Test the multiples of 20 to find the first one divisible by 8, 10, and 15.

Multiples: 20, 40, 60, 80, 100, 120.

Check: 120 ÷ 8 = 15, 120 ÷ 10 = 12, 120 ÷ 15 = 8. All are exact divisors.

∴ The correct answer is 120.

Alternate Method

Given: Numbers are 8, 10, 15, and 20.

Formula: The least number exactly divisible by a set of numbers is their Least Common Multiple (LCM).

Calculations:

⇒ Prime factorization of 8 = 2 × 2 × 2 = 2³

⇒ Prime factorization of 10 = 2 × 5 = 2¹ × 5¹

⇒ Prime factorization of 15 = 3 × 5 = 3¹ × 5¹

⇒ Prime factorization of 20 = 2 × 2 × 5 = 2² × 5¹

⇒ LCM = Product of highest powers of all prime factors

⇒ LCM = 2³ × 3¹ × 5¹

⇒ LCM = 8 × 3 × 5 = 120

∴ The correct answer is 120.

Additional Information

HCF and LCM Relation

For any two numbers a and b, the product of their HCF and LCM is equal to the product of the numbers: HCF(a, b) × LCM(a, b) = a × b.

LCM of Fractions

To find the LCM of fractions, use the formula: LCM of Numerators ÷ HCF of Denominators.

Co-prime Numbers

If two numbers are co-prime (their HCF is 1), then their LCM is always the product of the two numbers.